fix more math

src/8_differential_equations_2.md

@@ -489,12 +489,12 @@ the eigenfunctions of $L$.
     
     Recall that q quantum state $|\phi\rangle$ can be written in an orthonormal 
     basis $\{ |u_n\rangle \}$ as 
-    $$\|\phi\rangle = \underset{n}{\Sigma} \langle u_n | \phi \rangle\, |u_n\rangle.$$ 
+    $$|\phi\rangle = \underset{n}{\Sigma} \langle u_n | \phi \rangle\, |u_n\rangle.$$ 
     
     In terms of hermitian operators and their eigenfunctions, the eigenfunctions
     play the role of the orthonormal basis. In reference to our running example,
     the 1D Schrödinger equation of a free particle, the eigenfunctions 
-    $\sin(\frac{n \pi x}{L})$ play the role of the basis functions $\ket{u_n}$.
+    $\sin(\frac{n \pi x}{L})$ play the role of the basis functions $|u_n\rangle$.
     
 To close our running example, consider the initial condition 
 $\psi(x,o) = \psi_{0}(x)$. Since the eigenfunctions $\sin(\frac{n \pi x}{L})$ 
@@ -583,7 +583,7 @@ necessary to work with numerical methods of solution.
 6.  [:sweat:] We consider the Hilbert space of functions $f(x)$ defined
     for $x \ \epsilon \ [0,L]$ with $f(0)=f(L)=0$. 
 
-    Which of the following operators on this space is hermitian?
+    Which of the following operators on this space is Hermitian?
 
     (a) $L = A(x) \frac{d^2 f}{dx^2}$